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Next-generation surveys like the Legacy Survey of Space and Time (LSST) on the Vera C. Rubin Observatory will generate orders of magnitude more discoveries of transients and variable stars than previous surveys. To prepare for this data deluge, we developed the Photometric LSST Astronomical Time-series Classification Challenge (PLAsTiCC), a competition which aimed to catalyze the development of robust classifiers under LSST-like conditions of a non-representative training set for a large photometric test set of imbalanced classes. Over 1,000 teams participated in PLAsTiCC, which was hosted in the Kaggle data science competition platform between Sep 28, 2018 and Dec 17, 2018, ultimately identifying three winners in February 2019. Participants produced classifiers employing a diverse set of machine learning techniques including hybrid combinations and ensemble averages of a range of approaches, among them boosted decision trees, neural networks, and multi-layer perceptrons. The strong performance of the top three classifiers on Type Ia supernovae and kilonovae represent a major improvement over the current state-of-the-art within astronomy. This paper summarizes the most promising methods and evaluates their results in detail, highlighting future directions both for classifier development and simulation needs for a next generation PLAsTiCC data set.